1. Can a prince's rescuer run with him?
We say "yes", since the rules don't specify that he cannot run. Also, note that when he is in random movement he moves d6, which exceeds his assigned movement stat of 4, thus implying that he can run.
2. Does the winning player get the loot if all other warbands route?
2a: What if the prince is not in the winner's custody?
With the scenario as written, the game doesn't end when the other warbands route, only when a warband escorts the prince off of the board. Here's the rule: "The game ends when one warband manages to get the merchantís son off the table. That warband (and any allied warband) is the winner." So when a warband escorts the prince off the table, regardless of the status of the other warbands, that warband wins and it receives the loot.
3. Suppose one model wishes to charge two enemy models. One of the enemy models is a base-width closer to the charger. Can he intercept so that he's the only one being charged?
Here's the rule: "If an unengaged (ie, not in hand-to-hand combat) enemy model lies within 2" of the charge route, that model may choose to intercept the charger if he wishes." So, as I understand your question, yes the closer model could intercept.
4. How does the underdog bonus work in multiplayer? I seem to remember it being only the lowest rated band that receives the bonus xp...?
You've already answered this one.